?

-+++++

+ + ++

-+++++

+ + ++

+++

+++ +++ + -

+ ++ ++ ++ +

EXPLORE ACTIVITY

ESSENTIAL QUESTION

Modeling an Equation with a Variable on Both Sides Algebra tiles can model equations with a variable on both sides.

Use algebra tiles to model and solve x + 5 = 3x - 1.

How can you represent and solve equations with the variable on both sides?

L E S S O N

11.1Equations with the Variable on Both Sides

The solution is = .

KEY

--

= -1

= 1

+ = 0

= x

Reflect1. How can you check the solution to x + 5 = 3x - 1 using algebra tiles?

Math TalkMathematical Processes

8.8.C

Expressions, equations, and relationships—8.8.A Write one-variable equations or inequalities with variables on both sides that represent problems using rational number coefficients and constants. Also 8.8.B, 8.8.C

Why is a positive unit tile added to both sides

in the third step?

Model x + 5 on the left side of the mat and 3x - 1 on the right side.Remember that 3x - 1 is the same as

3x + .

Remove one x-tile from both sides. This

represents subtracting from both sides of the equation.

Place one +1-tile on both sides. This

represents adding to both sides of the equation. Remove zero pairs.

Separate each side into 2 equal groups.

One x-tile is equivalent to +1-tiles.

297Lesson 11.1

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Math TalkMathematical Processes

Math On the Spotmy.hrw.com

Solving an Equation with the Variable on Both SidesEquations with the variable on both sides can be used to compare costs of

real-world situations. To solve these equations, use inverse operations to get

the variable terms on one side of the equation.

Andy’s Rental Car charges an initial fee of $20 plus an additional $30 per

day to rent a car. Buddy’s Rental Car charges an initial fee of $36 plus an

additional $28 per day. For what number of days is the total cost charged

by the companies the same?

Write an expression representing the total cost of renting a car

from Andy’s Rental Car.

Initial fee + Cost for x days

20 + 30x

Write an expression representing the total cost of renting a car

from Buddy’s Rental Car.

Initial fee + Cost for x days

36 + 28x

Write an equation that can be solved to find the number of days

for which the total cost charged by the companies would be

the same.

Total cost at Andy’s = Total cost at Buddy’s

20 + 30x = 36 + 28x

Solve the equation for x.

The total cost is the same if the rental is for 8 days.

EXAMPLE 1

STEP 1

STEP 2

STEP 3

STEP 4

20 + 30x = 36 + 28x  -28x - 28x 20 + 2x = 36

 -20 -20

2x = 16

2x ___ 2

= 16 ___ 2

x = 8

Divide both sides by 2.

Subtract 28x from both sides.

Subtract 20 from both sides.

Animated Math

my.hrw.com

Write the equation.

8.8.A, 8.8.C

When is it more economical to rent from Andy’s Rental

Car? When is it more economical to rent

from Buddy’s?

Unit 4298

© Houghton Miff

lin Harcourt Pub

lishing

Company

• Image Credits:

©Roderic

k Chen/

Getty

Images

My Notes

Math Trainer

Online Assessment and Intervention

Personal

my.hrw.com

Math Trainer

Online Assessment and Intervention

Personal

my.hrw.com

Math On the Spot

my.hrw.com

Writing a Real-World Situation from an EquationAs shown in Example 1, an equation with the variable on both sides can be used

to represent a real-world situation. You can reverse this process by writing a

real-world situation for a given equation.

Write a real-world situation that could be modeled by the equation

150 + 25x = 55x.

The left side of the equation consists of a constant plus a variable

term. It could represent the total cost for doing a job where there

is an initial fee plus an hourly charge.

The right side of the equation consists of a variable term. It could

represent the cost for doing the same job based on an hourly

charge only.

The equation 150 + 25x = 55x could be represented by this situation:

A handyman charges $150 plus $25 per hour for house painting.

A painter charges $55 per hour. How many hours would a job have to

take for the handyman’s fee and the painter’s fee to be the same?

EXAMPLEXAMPLE 2

STEP 1

STEP 2

STEP 3

2. A water tank holds 256 gallons but is leaking at a rate of 3 gallons

per week. A second water tank holds 384 gallons but is leaking at

a rate of 5 gallons per week. After how many weeks will the amount

of water in the two tanks be the same?

YOUR TURN

3. Write a real-world situation that could be modeled by the equation

30x = 48 + 22x.

YOUR TURN

8.8.B

299Lesson 11.1

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Guided Practice

Use algebra tiles to model and solve each equation. (Explore Activity)

1. x + 4 = -x - 4 2. 2 - 3x = -x - 8

3. At Silver Gym, membership is $25 per month, and personal training

sessions are $30 each. At Fit Factor, membership is $65 per month,

and personal training sessions are $20 each. In one month, how many

personal training sessions would Sarah have to buy to make the total cost

at the two gyms equal? (Example 1)

4. Write a real-world situation that could be modeled by the equation

120 + 25x = 45x. (Example 2)

5. Write a real-world situation that could be modeled by the equation

100 - 6x = 160 - 10x. (Example 2)

6. How can you solve an equation with the variable on both sides?

ESSENTIAL QUESTION CHECK-IN??

Unit 4300

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

How can you solve equations with rational number coefficients and constants?

Math On the Spot

my.hrw.com

Math TalkMathematical Processes

?

Solving an Equation that Involves Fractions To solve an equation with the variable on both sides that involves fractions,

start by eliminating the fractions from the equation.

Solve 7 ___

10 n +  3 __

2 = 3 __

5 n + 2.

Determine the least common

multiple of the denominators: LCM(10, 5, 2) = 10

Multiply both sides of the equation by the LCM.

Use inverse operations to solve the equation.

Reflect1. What is the advantage of multiplying both sides of the equation by the

least common multiple of the denominators in the first step?

2. What If? What happens in the first step if you multiply both sides by a

common multiple of the denominators that is not the LCM?

EXAMPLEXAMPLE 1

STEP 1

STEP 2

STEP 3

7n + 15

- 15

7n -6n n

6n + 5

-6n 5

6n + 20

- 15 =

=

=

1

10 ( 7

__ 10

n + 3

_ 2

) = 10 ( 3

_ 5

n + 2 )

10 ( 7

__ 10

n ) + 10 ( 3

_ 2

) = 10 ( 3

_ 5

n ) + 10(2)

7n + 15 = 6n + 20

1 1 1

5 2

L E S S O N

11.2Equations with Rational Numbers

ESSENTIAL QUESTION

8.8.C

The constant on the right side, 2, is not a fraction. Why do you still need to

multiply it by the LCM, 10?

Expressions, equations, and relationships—8.8.A Write one-variable equations … with variables on both sides… using rational number coefficients and constants. Also 8.8.B, 8.8.C

10 = 10 × 1 = 5 × 2 = 2 × 5

Subtract 15 from both sides.

Subtract 6n from both sides.

303Lesson 11.2

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Math Trainer

Online Assessment and Intervention

Personal

my.hrw.com

Math On the Spotmy.hrw.com

Solve.

3. 1 _ 7

k - 6 = 3

_ 7

k + 4 4. 5 _ 6

y + 1 = -  1

_ 2

y + 1

_ 4

YOUR TURN

Solving an Equation that Involves DecimalsSolving an equation with the variable on both sides that involves decimals is

similar to solving an equation with fractions. But instead of first multiplying

both sides by the LCM, multiply by a power of 10 to eliminate the decimals.

Javier walks from his house to the zoo at a constant rate. After walking

0.75 mile, he meets his brother, Raul, and they continue walking at the

same constant rate. When they arrive at the zoo, Javier has walked for

0.5 hour and Raul has walked for 0.2 hour. What is the rate in miles

per hour at which the brothers walked to the zoo?

Write an equation for the distance from the brothers’ house to the

zoo, using the fact that distance equals rate times time. Let r = the

brothers’ walking rate.

distance to zoo = distance to zoo

0.2r + 0.75 = 0.5r

Multiply both sides of the equation

by 102 = 100.

EXAMPLE 2

STEP 1

STEP 2

100 ( 0.2 r ) + 100 ( 0.75 ) = 100 ( 0.5 r )

20 r + 75 = 50 r

Use inverse operations to solve the equation.

=

=

 2.5 =  r

20r + 75 - 20r 75 =

50r

-20r 30r

75

30 30r

30

So, the brothers’ constant rate of speed was 2.5 miles per hour.

STEP 3

8.8.A, 8.8.C

Multiplying by 100 clears the equation of decimals. Multiplying by 10 does not: 10 × 0.75 = 7.5.

Write the equation.Subtract 20r from both sides.

Divide both sides by 30.

Unit 4304

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

My Notes

Math Trainer

Online Assessment and Intervention

Personal

my.hrw.com

Math On the Spot

my.hrw.com

Math Trainer

Online Assessment and Intervention

Personal

my.hrw.com

5. Logan has two aquariums. One aquarium contains 1.3 cubic feet of

water and the other contains 1.9 cubic feet of water. The water in the

larger aquarium weighs 37.44 pounds more than the water in the

smaller aquarium. Write an equation with a variable on both sides to

represent the situation. Then find the weight of 1 cubic foot of water.

YOUR TURN

Writing a Real-World Situation from an EquationReal-world situations can often be represented by equations involving fractions

and decimals. Fractions and decimals can represent quantities such as weight,

volume, capacity, time, and temperature. Decimals can also be used to

represent dollars and cents.

Write a real-world situation that can be modeled by the equation

0.95x = 0.55x + 60.

The left side of the equation consists of a variable term. It could represent the

total cost for x items.

The right side of the equation consists of a variable term plus a constant. It

could represent the total cost for x items plus a flat fee.

The equation 0.95x = 0.55x + 60 could be represented by this situation: Toony

Tunes charges $0.95 for each song you download. Up With Downloads charges

$0.55 for each song but also charges an annual membership fee of $60. How

many songs must a customer download in a year so that the cost will be the

same at both websites?

EXAMPLEXAMPLE 3

6. Write a real-world problem that can be modeled by the equation

1 __ 3

x + 10 = 3 __ 5

x .

YOUR TURN

8.8.B

305Lesson 11.2

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

8. Write a real-world problem that can be modeled by the equation

1.25x = 0.75x + 50. (Example 3)

Guided Practice

1. Sandy is upgrading her Internet service. Fast Internet charges $60 for

installation and $50.45 per month. Quick Internet has free installation but

charges $57.95 per month. (Example 2)

a. Write an equation that can be used to find the number of months

after which the Internet service would cost the same.

b. Solve the equation.

Solve. (Examples 1 and 2)

2. 3 _

4 n - 18 = 1 _

4 n - 4 3. 6 + 4 _

5 b = 9 __

10 b

4. 2 __ 11

m + 16 = 4 + 6 __ 11

m

5. 2.25t + 5 = 13.5t + 14 6. 3.6w = 1.6w + 24 7. -0.75p - 2 = 0.25p

9. How does the method for solving equations with fractional or decimal

coefficients and constants compare with the method for solving

equations with integer coefficients and constants?

ESSENTIAL QUESTION CHECK-IN??

Unit 4306

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany